On the Multilinear Extensions of the Concept of Absolutely Summing Operators

نویسنده

  • DANIEL M. PELLEGRINO
چکیده

The core of the theory of absolutely summing operators lie in the ideas of A. Grothendieck in the 1950s. Further work (after a decade) of A. Pietsch [19] and Lindenstrauss and Pe l czyński [9] clarified Grothendiecks insights and nowadays the ideal of absolutely summing operators is a central topic of investigation. For details on absolutely summing operators we refer to the book by Diestel-JarchowTonge [7]. A natural question is how to extend the concept of absolutely summing operators to multilinear mappings and polynomials. A first light in this direction is the work by Alencar-Matos [1], where several classes of multilinear mappings between Banach spaces were investigated. Since then, just concerning to the idea of lifting the ideal of absolutely summing operators to polynomials and multilinear mappings, there are several works in different directions (we mention Bombal et al [2], Dimant [8], Matos [10],[11],[12]). However, there seems to be no effort in the direction of comparing these different classes. The aim of this paper is to investigate these classes and their connections. Throughout this paper E,E1, ..., En, G1, ..., Gn,F, F0 will be Banach spaces. Given a natural number n ≥ 2, the Banach space of all continuous n-linear mappings from E1 × ...× En into F endowed with the sup norm will be denoted by L(E1, ..., En;F ) and the space of all continuous n-homogeneous polynomials P from E into F with the sup norm is represented by P(E;F ). If T is a multilinear

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تاریخ انتشار 2003